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Properties

Label 2-325-13.4-c1-0-7
Degree $2$
Conductor $325$
Sign $-0.252 - 0.967i$
Analytic cond. $2.59513$
Root an. cond. $1.61094$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.89 + 1.09i)2-s + (−0.5 + 0.866i)3-s + (1.39 + 2.41i)4-s + (−1.89 + 1.09i)6-s + (−1.5 + 0.866i)7-s + 1.73i·8-s + (1 + 1.73i)9-s + (2.29 + 1.32i)11-s − 2.79·12-s + (−1 + 3.46i)13-s − 3.79·14-s + (0.895 − 1.55i)16-s + (−2.29 − 3.96i)17-s + 4.37i·18-s + (1.5 − 0.866i)19-s + ⋯
L(s)  = 1  + (1.34 + 0.773i)2-s + (−0.288 + 0.499i)3-s + (0.697 + 1.20i)4-s + (−0.773 + 0.446i)6-s + (−0.566 + 0.327i)7-s + 0.612i·8-s + (0.333 + 0.577i)9-s + (0.690 + 0.398i)11-s − 0.805·12-s + (−0.277 + 0.960i)13-s − 1.01·14-s + (0.223 − 0.387i)16-s + (−0.555 − 0.962i)17-s + 1.03i·18-s + (0.344 − 0.198i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.252 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $-0.252 - 0.967i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ -0.252 - 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42502 + 1.84485i\)
\(L(\frac12)\) \(\approx\) \(1.42502 + 1.84485i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 + (1 - 3.46i)T \)
good2 \( 1 + (-1.89 - 1.09i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (0.5 - 0.866i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (1.5 - 0.866i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.29 - 1.32i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.29 + 3.96i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.5 + 0.866i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.29 + 3.96i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.29 + 3.96i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 9.66iT - 31T^{2} \)
37 \( 1 + (-6.87 - 3.96i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.29 + 1.32i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.708 + 1.22i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 8.75iT - 47T^{2} \)
53 \( 1 + 1.58T + 53T^{2} \)
59 \( 1 + (-2.91 + 1.68i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.29 - 9.16i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (12.8 + 7.43i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.08 - 1.77i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 6T + 79T^{2} \)
83 \( 1 + 11.3iT - 83T^{2} \)
89 \( 1 + (3.70 + 2.14i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.87 + 2.23i)T + (48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.97387849940462873845815874447, −11.35197723284907706685286184827, −9.896667925780629804186978014448, −9.238271985888426153310402988148, −7.62004421921352067170964512435, −6.74404776009806882788166735013, −5.93994624560785862265356508317, −4.66291095058155772363953286685, −4.28274589033271267660801127895, −2.66257937384224182243757120324, 1.37277825334865716856496814071, 3.14910051132875279422328253182, 3.90678491358934518309408138495, 5.26841311765211245210416347601, 6.22988354968613841483745737696, 7.06115056004437633628981360395, 8.539906117571922625388071821822, 9.854332240901221920902920579878, 10.75124978222256162900707369695, 11.63137120791725017447186829873

Graph of the $Z$-function along the critical line